Set-Valued Kurzweil Henstock Pettis Integral

Transcription

1 Set-Vlued Anlysis (25) 13: Springer 25 Set-Vlued Kurzweil Henstock Pettis Integrl L. DI PIAZZA 1 nd K. MUSIAŁ 2 1 Deprtment of Mthemtics, University of Plermo, vi Archirfi 34, 9123 Plermo, Itly. e-mil: dipizz@mth.unip.it 2 Institute of Mthemtics, Wrocłw University, Pl. Grunwldzki 2/4, Wrocłw, Polnd. e-mil: musil@mth.uni.wroc.pl (Received: 17 Februry 23; in finl form: 11 November 23) Abstrct. It is shown tht the obvious generliztion of the Pettis integrl of multifunction obtined by replcing the Lebesgue integrbility of the support functions by the Kurzweil Henstock integrbility, produces n integrl which cn be described in cse of multifunctions with (wekly) compct convex vlues in terms of the Pettis set-vlued integrl. Mthemtics Subject Clssifictions (2): Primry: 28B2; secondry: 26A39, 28B5, 46G1, 54C6. Key words: multifunction, Pettis set-vlued integrl, Kurzweil Henstock integrl, Kurzweil Henstock Pettis integrl, support function, selection. 1. Introduction There is bundnt literture deling with Bochner nd Pettis integrtion of Bnch spce-vlued multifunctions (see El Amri nd Hess [5] for further references) of severl types. In prticulr, quite recently Zit [17, 18] nd El Amri nd Hess [5] presented nice chrcteriztion of Pettis integrble multifunctions hving s their vlues convex wekly compct subsets of Bnch spce. The definitions of such integrls involve in some wy the Lebesgue integrbility of the support functions. The theory of integrtion introduced by Lebesgue t the beginning of the twentieth century is powerful tool which, perhps becuse of its bstrct chrcter, does not hve the intuitive ppel of the Riemnn integrl. Besides, s Lebesgue himself observed in his thesis [15], his integrl does not integrte ll unbounded derivtives nd so it does not provide solution for the problem of primitives, i.e. for the problem of recovering function from its derivtive. Moreover the Lebesgue theory does not cover nonbsolutely convergent integrls. In 1957 Kurzweil [14] nd, independently, in 1963 Henstock [9], by simple modifiction of Riemnn s method, gve new definition of integrl, which is more generl thn tht of Lebesgue. The Kurzweil Henstock integrl retins the intuitive ppel of the Riemnn definition, nd, t the sme time, hs the power of Lebesgue s one. Moreover it integrtes ll derivtives. In the lst thirty yers the theory of nonbsolute integrls hs gone on considerbly nd the reserches in this

2 168 L. DI PIAZZA AND K. MUSIAŁ field re still ctive nd fr to be complete (for survey on the subject, we refer to [1]). This is the motivtion to consider, lso in cse of multifunctions, the Kurzweil Henstock integrl for rel vlued functions. In this pper, in prticulr, we study the obvious generliztion of the Pettis integrl of multifunction obtined by replcing the Lebesgue integrbility of the support functions by the Kurzweil Henstock integrbility (we cll such n integrl Kurzweil Henstock Pettis). In Theorem 1 we prove surprising nd unexpected chrcteriztion of the new integrl in terms of the Pettis integrl: the Kurzweil Henstock Pettis integrl is in some wy trnsltion of the Pettis integrl. In cse of mesurble multifunctions with convex wekly compct vlues, the Pettis integrbility of the selections is necessry nd sufficient condition for the Pettis integrbility of the multifunction. We show tht similr chrcteriztion holds true lso in cse of the Kurzweil Henstock Pettis integrbility of multifunctions. 2. Bsic Fcts Let [, 1] be the unit intervl of the rel line equipped with the usul topology nd the Lebesgue mesure. L denotes the fmily of ll Lebesgue mesurble subsets of [, 1] nd if E L, then E denotes its Lebesgue mesure. A prtition P in [,b] [, 1] is collection {(I 1,t 1 ),...,(I p,t p )}, wherei 1,...,I p re nonoverlpping subintervls of [,b] nd t i is point of I i, i = 1,...,p.If p i=1 I i =[,b], we sy tht P is prtition of [,b]. Aguge on [,b] is positive function on [,b]. For given guge δ on [,b], we sy tht prtition {(I 1,t 1 ),...,(I p,t p )} is δ-fine if I i (t i δ(x i ), t i + δ(x i )), i = 1,...,p. DEFINITION 1 ([3, 6]). Let X be ny Bnch spce. A function f : [, 1] X is sid to be Henstock integrble on [,b] [, 1] if there exists w X with the following property: for every ɛ>there exists guge δ on [,b] such tht p f(t i ) I i w <ε, i=1 for ech δ-fine prtition {(I 1,t 1 ),...,(I p,t p )} of [,b].wesetw =: (H ) b f dt. We denote the set of ll Henstock integrble functions on [, 1], tking their vlues in X,byH([, 1],X). In cse when X is the rel line, f is clled Kurzweil Henstock integrble, orsimplykh-integrble nd the spce of ll KH-integrble functions is denoted by KH[, 1]. It is useful to recll some bsic results in the theory of rel vlued KH-integrble functions. The proofs cn be found, for exmple, in [8]. PROPOSITION 1. Let f : [, 1] R be function.

3 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 169 () If f is Lebesgue integrble on [, 1], then it is lso KH-integrble. (b) If f is KH-integrble on [, 1],thenf is mesurble. (c) If f is KH-integrble on [, 1], thenf is KH-integrble on every subintervl of [, 1]. (d) f is Lebesgue integrble on [, 1] if nd only if both f nd f re KHintegrble. (e) If f = F is derivtive, then f is KH-integrble nd (KH) s f(t)dt = F(s) F(),forechs [, 1]. Throughout this pper X is seprble Bnch spce with dul X. The closed unit bll of X is denoted by B(X ). c(x) denotes the collection of ll nonempty closed convex subsets of X. cwk(x) (resp. cwk(x )) denotes the fmily of ll nonempty convex wekly compct subsets of X (resp. of the bidul X of X), ck(x) (resp. ck(x )) the fmily of ll nonempty convex compct subsets of X (resp. of X )ndcb(x) (resp. cb(x )) the fmily of ll nonempty closed bounded convex subsets of X (resp. of X ). For every C c(x) the support function of C is denoted by s(,c)nd defined on X by s(x,c) = sup{ x,x :x C}, for ech x X.IfC =, s(,c)is identiclly.otherwises(,c)does not tke the vlue. Any mp Γ : [, 1] c(x) is clled multifunction. A multifunction Γ is sid to be mesurble if for ech open subset O of X, the set {t [, 1] :Γ(t) O }is mesurble set. Γ is sid to be sclrly mesurble if for every x X,themps(x,Γ( )) is mesurble. It is known tht in cse of cwk(x)-vlued multifunctions the sclr mesurbility yields the mesurbility (cf. [1], Proposition 2.39). The reverse impliction lwys holds true (cf. [1], Proposition 2.3.2). Γ : [, 1] c(x) is sid to be grph mesurble if the set {(t, x) [, 1] X : x Γ(t)} is member of the product σ -lgebr generted by L nd the Borel subsets of X in the norm topology. In cse of considering of complete probbility spce nd Bnch spce the grph mesurbility of c(x)-vlued multifunction coincides with its mesurbility (cf. [1], Theorem ). Γ is sid to be sclrly integrble (resp. sclrly Kurzweil Henstock integrble) if, for every x X, the function s(x,γ( )) is integrble (resp. Kurzweil Henstock integrble). A function f : [, 1] X is clled selection of Γ if, for every t [, 1], one hs f(t) Γ(t). A selection f is sid to be mesurble if the function f is strongly mesurble (i.e. f is limit of n lmost everywhere convergent sequence of mesurble simple functions). DEFINITION 2. A mesurble multifunction Γ : [, 1] cb(x) is Dunford (respectively Kurzweil Henstock Dunford) integrble or simply D-integrble (resp. KHD-integrble), if it is sclrly integrble (resp. sclrly Kurzweil Henstock integrble) nd for ech nonempty set A L (resp. subintervl [,b] [, 1]), there exists nonempty set W A cb(x ) (resp. W [,b] cb(x )) such tht

4 17 L. DI PIAZZA AND K. MUSIAŁ s(x,w A ) = (L) ( resp. s(x,w [,b] ) = (KH) A s(x, Γ (t)) dt, (1) b ) s(x, Γ (t)) dt for ll x X (L stnds for the Lebesgue integrl). If W A C (resp. W [,b] C), for ech A L (resp. [,b] [, 1]), nd C is subspce of cb(x), thenγ is sid to be Pettis integrble, or simply P -integrble (resp. Kurzweil Henstock Pettis integrble or simply KHP-integrble)inC. We cll W A (resp. W [,b] )thepettis (resp. Kurzweil Henstock Pettis) integrl of Γ over A (resp. [,b])ndwesetw A =: (P ) A Γ(t)dt (resp. W [,b] =: (KHP) b Γ(t)dt). We note tht when multifunction is function f : [, 1] X, then the sets W A nd W [,b] re reduced to vectors in X, the equlities (1) nd (2) turn into x,w A =(L) x f(t)dt, ( resp. x,w [,b] =(KH) A b ) x f(t)dt nd we sy in tht cse tht the function f is Pettis (resp. Kurzweil Henstock Pettis) integrble. It is perhps worth to recll in this plce tht Gmez nd Mendoz [7] proved tht function f : [, 1] X is KHD-integrble if nd only if f is sclrly KH-integrble. An extensive study of Bnch vlued Pettis integrl cn be found in [16]. Given multifunction Γ : [, 1] cb(x) by the symbols S KHP (Γ ) nd S P (Γ ) we denote the fmilies of ll mesurble selections of Γ tht re respectively Kurzweil Henstock Pettis integrble nd Pettis-integrble. It is consequence of [13] tht if X is seprble, then for ech mesurble multifunction Γ : [, 1] cb(x) the fmily of mesurble selections of Γ is not empty. DEFINITION 3. A mesurble multifunction Γ : [, 1] cwk(x) is sid to be Aumnn Kurzweil Henstock Pettis integrble if S KHP (Γ ). Then we define 1 { 1 } (AKHP) Γ(t)dt := (KHP) f(t)dt : f S KHP (Γ ). It is cler tht ech Henstock integrble function is lso KHP-integrble. The reverse impliction is not so obvious. In [7] there is n exmple of KHP-integrble function f : [, 1] c (the uthors sy there on the Denjoy Pettis integrl) which is not Pettis integrble. We re going to show tht sme function is not Henstock integrble. It will follow from this tht the collection of KHP-integrble functions is lrger thn tht of Henstock integrble ones. (2)

5 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 171 EXAMPLE 1. Consider sequence of intervls A n =[ n,b n ] [, 1] such tht 1 =, b n < n+1 for ll n N nd lim n b n = 1 nd define f : [, 1] c by ( 1 f(t)= 2 A 2n 1 χ A 2n 1 (t) 1 ) 2 A 2n χ A 2n (t). n=1 The function f is Dunford integrble, (D) [,1] f = nd(d) J f belongs to c for ech subintervl J [, 1] (see [7]). Consequently, f is KHP-integrble in [, 1] nd (KHP) [,1] f =. Let us consider now ny guge δ on [, 1].SetI =[c, 1],wherec>1 δ(1) nd b 2n 1 <c< 2n, for suitble nturl number n. Then, for i = 1,..., 2n 1, let P i ={(Is i,yi s ) : s = 1,...,p i} be δ-fine prtition of [ i,b i ] nd let Pˆ i = {(Js i,zi s ) : s = 1,...,q i} be δ-fine prtition of [b i, i+1 ],fori = 1,...,2n 2, nd of [b 2n 1,c] for i = 2n 1. Moreover ssume tht if zs i = b i or, respectively, zs i = i+1for some indices i nd s then the corresponding intervl Js i stisfies the dditionl condition Js i < 1 4(2n 1) A i or, respectively, Js i < 1 4(2n 1) A i+1. By the construction the fmily P ={(I, 1)} ( i P i) ( ˆ i P i )} is δ-fine prtition of [, 1].Wehve f(t) I (I,t) P = f(1) I + + i=1 1 2 {i:z i s = i+1} 2n 1 p i s=1 {i:z i s =b i} 2n 1 i=1 p i s=1 f(zs i ) J s i f(ys i ) I s i Js i 2 A i f(y i s ) I i s + {i:z i s =b i} {i:z i s = i+1} {i:z i s =b i} f(z i s ) J i s + f(zs i ) J s i J i s 2 A i+1 {i:z i s = i+1} f(z i s ) J i s 1 2 (2n 1 1) 8(2n 1) (2n 1 1) 8(2n 1) = 1 4. So tking into ccount tht (D) [,1] f =, the inequlities show tht f cnnot be H -integrble. It seems to be good plce to put here remrk concerning the problem of primitives for Bnch spce vlued functions. PROPOSITION 2. If f : [, 1] X is wekly differentible, then its wek derivtive f is KHP-integrble nd (KHP) s f (t) dt = f(s) f() for ech s [, 1].

6 172 L. DI PIAZZA AND K. MUSIAŁ Proof. The existence of the wek derivtive t point t mens tht there is point f (t) X such tht x f(t + t) x f(t) lim = x f (t) t t for ech x X. This in prticulr mens tht ech x f is differentible nd so by (e) of Proposition 1, x f(s) x f() = (KH) s (x f) (t) dt for every s [, 1]. But by the ssumption, we hve (x f) = x f wht yields x f(s) x f() = (KH) s x f (t) dt nd mens exctly tht f(s) f() = (KHP) s f (t) dt. In the bove proof one my ssume the wek continuity of f everywhere nd the wek differentibility of f nerly everywhere, i.e. except for countble set (cf. [8]). 3. A Chrcteriztion of KHP-Integrble Multifunctions We begin with n esy fct (it is true in more generl cse of c(x) insted of cb(x), but we do not wnt to enter into detils concerning the definition of the Pettis integrl in such cse, see [5]). LEMMA 1. Let G: [, 1] cb(x) be Pettis integrble in cb(x). If the null function is selection of G, then for every mesurble sets A nd B such tht A B we hve W A W B. Proof. Suppose tht x W A \ W B. Then, due to the Hhn Bnch theorem, there is x such tht x (x )>sup x WB x (x). Consequently, (L) s(x, G(t)) dt = s(x,w A) x (x )> sup x (x) A x W B = s(x,w B) = (L) s(x, G(t)) dt B wht contrdicts the nonnegtivity of the support functions of G. Thus W A W B. LEMMA 2. If Γ : [, 1] cwk(x) is KHP-integrble, then ech mesurble selection of Γ is KHP-integrble. Proof. If f is mesurble selection of Γ, then for ech x X nd t [, 1] we hve the inequlity s( x, Γ (t)) x f(t) s(x, Γ (t)). (3)

7 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 173 So, if f is mesurble, then the Kurzweil Henstock integrbility of the function x f follows immeditely by the Kurzweil Henstock integrbility of s(x, Γ (t)), for ech x X. Indeed, we get from (3) the inequlities, x f(t)+ s( x, Γ (t)) s(x, Γ (t)) + s( x, Γ (t)). The function s(x,γ( )) + s( x,γ( )) is nonnegtive nd KH-integrble, hence it is Lebesgue integrble (see (d) of Proposition 1). Consequently lso x f( ) + s( x,γ( )) is Lebesgue integrble. Finlly x f(t)=[x f(t)+ s( x, Γ (t))] s( x, Γ (t)) nd so x f KH[, 1]. Hence for ech [,b] [, 1] we hve s( x,w [,b] ) (KH) b x f(t)dt s(x,w [,b] ). Since W [,b] is convex wekly compct, the function x s(x,w [,b] ) is τ(x,x)-continuous, where τ(x,x)is the Mckey topology of X. Consequently the functionl x (KH) b x f(t)dt is lso τ(x,x)-continuous. It follows tht there is x [,b] X such tht (KH) b x f(t)dt = x,x [,b]. LEMMA 3. If ll mesurble selections of mesurble multifunction Γ : [, 1] cwk(x) re KHP-integrble, then for every x X nd every [,b] [, 1], we hve s ( x,(akhp) b ) Γ(t)dt = (KH) b s(x, Γ (t)) dt. Proof. It is enough to prove the ssertion for the unit intervl. If f S KHP (Γ ), then x f(t) s(x, Γ (t)) nd so we get immeditely the inequlity ( 1 ) 1 s x,(akhp) Γ(t)dt (KH) s(x, Γ (t)) dt. To prove the reverse inequlity let us fix x nd ε>. Then set Γ ε (t) := Γ(t) {x X : x (x) s(x, Γ (t)) ε}. One cn esily see tht Γ ε is grph mesurble nd so it is mesurble (cf. [1], Theorem2.1.35).Iff is mesurble selection of Γ ε, then clerly x f(t) s(x, Γ (t)) ε nd further (KH) 1 x f(t)dt (KH ) 1 This completes the proof. s(x, Γ (t)) dt ε.

8 174 L. DI PIAZZA AND K. MUSIAŁ LEMMA 4. If ll mesurble selections of sclrly KH-integrble multifunction Γ : [, 1] cwk(x) re KHP-integrble, then (AKHP) 1 Γ(t)dt is convex wekly compct set. Proof. Notice first tht the sclr KH-integrbility of Γ yields the sclr mesurbility of Γ nd hence its mesurbility (becuse we consider cwk(x)-vlued multifunctions). Let us fix mesurble selection f of Γ nd let G(t) := Γ(t) f(t).bythe ssumption f is KHP-integrble nd so G is lso AKHP-integrble. Let I [,1] := (AKHP) 1 G(t) dt nd let D be countble wek -dense subset of B(X ).Itis enough to prove the convexity nd wek compctness of I [,1]. As the convexity is obvious we will try to prove the wek compctness of I [,1]. To do it tke sequence of points x n I [,1]. Then there exist g n S KHP (G) with x n = (KHP) 1 g n (t) dt. We hve for ech n N, ech t [, 1] nd ech x the inequlities s( x, G(t)) x g n (t) s(x, G(t)) (4) nd the support functions s(x, G(t)) re Lebesgue integrble (becuse they re nonnegtive nd KH-integrble). Consequently lso ech x g n is Lebesgue integrble nd (L) 1 x g n (t) dt (L) 1 s(x, G(t)) dt + (L) 1 s( x, G(t)) dt. Due to the countbility of D nd L 1 -boundedness of ech sequence x g n we cn pply Bukhvlov Loznovskij s theorem [2] to find h n conv{g n,g n+1,...} such tht for ech x D the sequence x h n is.e. convergent to mesurble function h x. (We could pply lso Komlos theorem [11] insted, but the proof of the result of Bukhvlov Loznovskij is much more elementry.) As for ech t nd n we hve h n (t) G(t) nd G(t) is wekly compct there is wek cluster point h(t) G(t) of h n (t). It follows tht there is set N of Lebesgue mesure zero such tht for ech x D nd ech t/ N we hve x h(t) = lim n x h n (t) = h x (t). But s D seprtes points of X nd h n (t) is wekly reltively compct it follows tht for ech t / N the sequence h n (t) is wekly convergent to h(t). The wek mesurbility of h is immedite nd s X is seprble the Pettis theorem yields its strong mesurbility. Moreover, s consequence of the KHP integrbility of ll mesurble selections of Γ nd of the KHP-integrbility of f,we hve h S KHP (G).

9 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 175 Tking into ccount (4) nd the Lebesgue dominted convergence theorem we get for ech x X the reltion 1 1 lim x,(khp) h n (t) dt = lim(l) x h n (t) dt n n 1 = (L) x h(t) dt 1 = x,(khp) h(t) dt. (5) Put now y n = (KHP) 1 h n(t) dt.theny n I [,1], y n conv{x n,x n+1,...} nd the sequence y n is wekly convergent to x = (KHP) 1 h(t) dt. Thus, given n rbitrry sequence of elements x n I [,1] there is convex combintion of points y n conv{x n,x n+1,...} nd x I [,1] such tht y n x wekly. Consequently, the set I [,1] is wekly compct (cf. [12], 24). THEOREM 1. Let Γ : [, 1] cwk(x) be sclrly Kurzweil Henstock integrble multifunction. Then the following conditions re equivlent: (i) Γ is KHP-integrble in cwk(x); (ii) S KHP (Γ ) nd for every f S KHP (Γ ) there exists multifunction G: [, 1] cwk(x) such tht Γ(t)= G(t) + f(t)nd G is Pettis integrble in cwk(x); (iii) there exists f S KHP (Γ ) nd multifunction G: [, 1] cwk(x) such tht Γ(t)= G(t) + f(t)nd G is Pettis integrble in cwk(x); (iv) S KHP (Γ ) nd for every f, h S KHP (Γ ), h f is Pettis integrble; (v) for ll [,b] [, 1], (AKHP) b Γ(t)dt belongs to cwk(x) nd ( b ) b s x,(akhp) Γ(t)dt = (KH) s(x, Γ (t)) dt for ll x X ; (vi) ech mesurble selection of Γ is KHP-integrble. Proof. (i) (ii) Let f S KHP (Γ ) be quite rbitrry (existing by Lemm 2). Define G: [, 1] cwk(x) by setting G(t) := Γ(t) f(t).thens(x, G(t)) for ll x nd t [, 1]. Moreover, s(x, Γ (t)) = s(x, G(t)) + x f(t), (6) nd so KH-integrbility of s(x,γ( )) nd of x f yields the Lebesgue integrbility of s(x,g( )). Thus, we hve for ech x X (L) 1 s(x, G(t)) dt = (KH) 1 s(x, Γ (t)) dt (KH) 1 x f(t)dt

10 176 L. DI PIAZZA AND K. MUSIAŁ 1 = s(x,w [,1] ) (KH) ( = s x,w [,1] (KHP) 1 x f(t)dt ) f(t)dt. And since W [,1] belongs to cwk(x),lsothesetw [,1] (KHP) 1 f(t)dt belongs to cwk(x). Since the zero function is Pettis integrble selection of G nd s(x,g( )) is Lebesgue integrble, it follows from Theorem 3.7 of [5] tht for every E L there is closed convex set C E X such tht (L) s(x, G(t)) dt = s(x,c E ). E But s the set W [,1] (KHP) 1 f(t)dt is wekly compct, we my pply Lemm 1 to get Pettis integrbility of G in cwk(x). (ii) (iv) Let f S KHP (Γ ) nd set G(t) := Γ(t) f(t).nowifh is mesurble selection of Γ then g = h f is mesurble selection of G nd we hve the inequlity s( x, G(t)) x g(t) s(x, G(t)). (7) Hence, for ech E L there is W E cwk(x) such tht s( x,w E ) = (L) s( x, G(t)) dt E (L) x g(t) dt E (L) s(x, G(t)) dt = s(x,w E ). E Since W E is wekly compct its support function is τ(x,x)-continuous. It follows tht x (L) E x g(t) dt is τ(x,x)-continuous. This implies the Pettis integrbility of g. (iv) (ii) Tke n f S KHP (Γ ). Then, by the ssumption, ech mesurble selection g of G = Γ f is Pettis integrble nd so, by Theorem 5.4 of [5], G is Pettis integrble in cwk(x). (ii) (v) Let f S KHP (Γ ) be such tht the multifunction G(t) = Γ(t) f(t) is Pettis integrble in cwk(x). According to Theorem 5.4 of [5] we hve (P ) 1 G(t) dt ={(P ) 1 f(t)dt : f S P (G)}. Since(P ) 1 G(t) dt is convex wekly compct set nd (AKHP) 1 Γ(t)dt = (P ) 1 G(t) dt + (KHP) 1 lso the set (AKHP) 1 Γ(t)dt is convex wekly compct. f(t)dt,

11 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 177 Now we prove the second prt of the ssertion. Its proof is similr to the proof of Lemm 3. We tke [,b] =[, 1] for simplicity nd fix x X.Sinces(x, Γ (t)) = s(x, G(t)) + x f(t), by the hypotheses the support function s(x,γ( )) is KH-integrble. Then, tking into ccount tht s(x, G(t)) (KH) 1 x f dt (KH) 1 s(x, Γ (t)) dt for ech f S KHP (Γ ). Hence, ( 1 ) s x,(akhp) Γ(t)dt (KH) 1 s(x, Γ (t)) dt. Tke now n rbitrry ε> nd define new multifunction H : [, 1] cwk(x) by setting for ech t [, 1] H(t) := Γ(t) {x X : x (x) s(x, Γ (t)) ε}. If h is mesurble selection of H, s(x, Γ (t)) ε x h(t) s(x, Γ (t)) nd so (KH) 1 x h dt (KH) 1 s(x, Γ (t)) dt ε. Since h is lso selection of Γ, the ssertion follows. (v) (i), (ii) (iii) nd (iii) (i) re obvious. By Lemm 1 (i) implies (vi) nd (vi) (i) follows from Lemmt 3 nd 4. This completes the proof. Remrk 1. Theorem 1 remins true if cwk(x) is replced by ck(x). The proof requires only obvious chnges. COROLLARY 1. If Γ, G nd f re s in Theorem 1, then the (KHP)-integrl of Γ is trnsltion of the Pettis integrl of G: (KHP) b for ll [,b] [, 1]. Γ(t)dt = (P ) b G(t) dt + (KHP) b f(t)dt Anlysis of the proof of Theorem 1 gives lso the following result: THEOREM 2. Let Γ : [, 1] cb(x) be mesurble nd sclrly KH-integrble multifunction. If S KHP (Γ ), then the following conditions re equivlent: (j) Γ is KHP-integrble in cb(x);

12 178 L. DI PIAZZA AND K. MUSIAŁ (jj) for every f S KHP (Γ ) the multifunction G: [, 1] cb(x), given by G(t) = Γ(t) f(t), is Pettis integrble in cb(x); (jjj) there exists f S KHP (Γ ) such tht the multifunction G: [, 1] cb(x), given by G(t) = Γ(t) f(t), is Pettis integrble in cb(x). We hve then (KHP) b for ll [,b] [, 1]. Γ(t)dt = (P ) b G(t) dt + (KHP) b f(t)dt Without the ssumption S KHP (Γ ) we get only the following: THEOREM 3. Let Γ : [, 1] cb(x) be mesurble nd sclrly KH-integrble multifunction. If Γ is KHP-integrble in cb(x), then for every mesurble selection f of Γ the multifunction G: [, 1] cb(x) given by G(t) = Γ(t) f(t) is Pettis integrble in cb(x). We hve then (KH) b s(x, Γ (t)) dt = (L) b s(x, G(t)) dt + (KH) b x f(t)dt for ll [,b] [, 1] nd ll x X. Proof. Let f be mesurble selection of Γ nd let G be defined in the wy described bove. As mesurble selection of sclrly KH-integrble multifunction is sclrly KH-integrble, we see tht Γ(t)= G(t)+f(t), wheref is sclrly KHintegrble. Since G hs t lest one Bochner integrble selection (the null function), ccording to Corollry 3.8 of [5], for every E L there is D E c(x) with s(x,d E ) = (L) s(x, G(t)) dt E for ll x X. Hence, we hve for every x X s(x,d [,1] ) + (KH) 1 x f(t)dt = (KH) 1 s(x, Γ (t)) dt ±. It follows tht s(x,d [,1] ) ± for ll x X. The Bnch Steinhus theorem yields now D [,1] cb(x). And so pplying Lemm 1 we get tht lso every D E is bounded, proving the Pettis integrbility of G in cb(x). The Theorems 1 nd 2 show tht under suitble ssumptions mesurble nd sclrly (KH)-integrble multifunction is sum of Pettis integrble multifunction nd selection. In cse of cwk(x)-vlued multifunction ll selections of the Pettis integrble component re Pettis integrble. We finish with n exmple showing tht decomposition theorem for Pettis integrble in cwk(x) multifunction G in the form G(t) = H(t) + f(t),where f S P (G) nd ll mesurble selections of H re Bochner integrble, is in generl flse. In prticulr H cnnot be tken integrbly bounded.

13 SET-VALUED KURZWEIL HENSTOCK PETTIS INTEGRAL 179 EXAMPLE 2. Let f : [, 1] X be Pettis but not Bochner integrble function (it is well known tht if X is infinite-dimensionl, then there re such functions). We define multifunction G: [, 1] ck(x) by setting G(t) = conv{,f(t)}. Since for every x X we hve s(x, G(t)) =[x f(t)] +, the multifunction G is sclrly integrble nd mesurble. Moreover ll mesurble selections of G re Pettis integrble nd so G itself is Pettis integrble in ck(x) (see [5]). Suppose there is h S P (G) which is not Bochner integrble nd H(t) := G(t) h(t) is such tht ll its mesurble selections re Bochner integrble. We hve H(t) = conv{,f(t)} h(t) = conv{ h(t), f (t) h(t)}. But the function h is non- Bochner integrble selection of H. References 1. Bongiorno, B.: The Henstock Kurzweil integrl, In: Hndbook of Mesure Theory, Vols. I, II, North-Hollnd, Amsterdm, 22, pp Bukhvlov, A. nd Loznovskij, G. J.: On sets closed in mesure in spces of mesurble functions, Trudy Moskov. Mt. Obshch. 34 (1977), (in Russin). 3. Co, S. S.: The Henstock integrl for Bnch-vlued functions, SEA Bull. Mth. 16 (1992), Diestel, J. nd Uhl, J. J.: Vector Mesures, Mth. Surveys 15, El Amri, K. nd Hess, C.: On the Pettis integrl of closed vlued multifunctions, Set-Vlued Anl. 8 (2), Fremlin, D. H.: The Henstock nd McShne integrls of vector-vlued functions, Illinois J. Mth. 38 (1994), Gmez, J. L. nd Mendoz, J.: On Denjoy Dunford nd Denjoy Pettis integrls, Studi Mth. 13 (1998), Gordon, R. A.: The Integrls of Lebesgue, Denjoy, Perron, nd Henstock, Grd. Stud. Mth. 4, Amer. Mth. Soc., Henstock, R.: Theory of Integrtion, Butterworths, London, Hu, S. nd Ppgeorgiou, N. S.: Hndbook of Multivlued Anlysis. Vol. I: Theory, Kluwer Acdemic Publishers, Komlos, J.: A generliztion of problem of Steinhus, Act Mth. Acd. Sc. Hungrice 18 (1967), Köthe, G.: Topologicl Vector Spces I, Springer-Verlg, Kurtowski, K. nd Ryll-Nrdzewski, C.: A generl theorem on selections, Bull. Acd. Polon. Sci. Sèr. Sci. Mth. Astronom. Phys. 13 (1965), Kurzweil, J.: Generlized ordinry differentil equtions nd continuous dependence on prmeter, Czechoslovk Mth. J. 7 (1957), Lebesgue, H., Integrl, longueur, mire, Annli Mt. Pur Applic. 7 (192), Musił, K.: Topics in the theory of Pettis integrtion, Rend. Istit. Mt. Univ. Trieste 23 (1991), Zit, H.: Convergence theorems for Pettis integrble multifunctions, Bull. Polish Acd. Sci. Mth. 45 (1997), Zit, H.: On chrcteriztion of Pettis integrble multifunctions, Bull. Polish Acd. Sci. Mth. 48 (2),